Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions

TL;DR

This paper introduces a delayed rejection variant of Hamiltonian Monte Carlo that adaptively reduces step size after rejections, significantly improving sampling efficiency for multiscale distributions and robustness to step size errors.

Contribution

It proposes a novel delayed rejection approach for HMC that adaptively adjusts step sizes, enhancing performance on multiscale and challenging distributions.

Findings

Up to five-fold increase in effective sample size per gradient evaluation.

Improved robustness to step size misspecification.

Effective sampling of multiscale distributions like Neal's funnel.

Abstract

The efficiency of Hamiltonian Monte Carlo (HMC) can suffer when sampling a distribution with a wide range of length scales, because the small step sizes needed for stability in high-curvature regions are inefficient elsewhere. To address this we present a delayed rejection variant: if an initial HMC trajectory is rejected, we make one or more subsequent proposals each using a step size geometrically smaller than the last. We extend the standard delayed rejection framework by allowing the probability of a retry to depend on the probability of accepting the previous proposal. We test the scheme in several sampling tasks, including multiscale model distributions such as Neal's funnel, and statistical applications. Delayed rejection enables up to five-fold performance gains over optimally-tuned HMC, as measured by effective sample size per gradient evaluation. Even for simpler distributions, delayed rejection provides increased robustness to step size misspecification. Along the way, we provide an accessible but rigorous review of detailed balance for HMC.